B. P. Lathi, Zhi Ding - Modern Digital and Analog Communication Systems-Oxford University Press (2009)

482 RANDOM PROCESSES AND SPECTRAL ANALYSIS
If x(t) and y(t) are uncorrelated, then from Eq. (9.34),
and
Rxy (r) = Ryx(r) = xy
Rz(r) = Rx(r) + Ry (r) + 2:x.y (9.43)
Most processes of interest in communication problems have zero means. If processes x(t) and
y(t) are uncorrelated with either x or y = 0 [i.e., if x(t) and y(t) are incoherent], then
Rz(r) = Rx (r) + Ry (r)
(9.44a)
and
(9.44b)
It also follows from Eqs. (9.44a) and (9.19) that
(9.44c)
Hence, the mean square of a sum of incoherent ( or orthogonal) processes is equal to the sum
of the mean squares of these processes.
Example 9 . 10 Two independent random voltage processes x1 (t) and x2 (t) are applied to an RC network, as
shown in Fig. 9. 14. It is given that
2a
S x 2 (j ) = a 2 + (2rrj )2
Fi g
ure 9.14
Noise
calculations in a
resistive circuit.
X2(t)
y(t)
Determine the PSD and the power P y
of the output random process y(t). Assume that the
resistors in the circuit contribute negligible thermal noise (i.e., assume that they are noiseless).
Because the network is linear, the output voltage y(t) can be expressed as
y(t) = YI (t) + y2(t)
where Y1 (t) is the output from input x1 (t) [assuming x2(t) = OJ and y2 (t) is the output
from input x2 (t) [assuming x1 (t) = OJ. The transfer functions relating y(t) to x1 (t) and
x2 (t) are H1 (f) and H2 (f), respectively, given by
1
Hi (f) =
3(3 - j2rr/ + I)'
H 2(f) =
1
2(3 · j2rr/ + 1)